This invention relates in general to travelling-wave optical modulators and more particularly to travelling-wave modulators having an electrode structure that increases the bandwidth-to-drive-voltage ratio over conventional travelling-wave optical modulators. The structure of conventional optical modulators is discussed in chapter 14 of the text by Amnon Yariv entitled Quantum Electronics, 2nd Edition, John Wiley & Sons, Inc., 1975. In such modulators, optically transparent materials are used that, for a given direction of transmission of light in the material, exhibit an ordinary index of refraction n.sub.o for a first polarization of the light and exhibit an extraordinary index of refraction n.sub.e for a second polarization that is perpendicular to the first one. At least one of these indices of refraction is changeable in response to an applied voltage. The index of refraction can be changed by the applied voltage, for example, via the electrooptic effect or the photoelastic effect.
Each of these polarizations functions as a separate channel for transmission of light. Because the phase velocity of each of these channels is equal to the speed of light c divided by the index of refraction for that channel, the phase velocities for these two channels will generally be unequal. Since the phase of light at the output of the modulator is equal to the input phase plus 2*pi*f*L/v (where f is the is the frequency of the light, L is the length of the light path in the modulator and v is the phase velocity of the light), these modulators can be used to modulate the output phase of light in at least one of these channels. For sufficiently small applied voltages, the variation of phase velocity as a function of applied field is substantially linear so that the phase modulation is proportional to the applied voltage.
Phase modulation can be converted to amplitude modulation by interference of the light in one of these channels with another beam of light, such as the beam of light in the other channel. In such an amplitude modulator, the light in these two channels can be combined by a polarizer placed at the output of the modulator and orientated in a direction midway between the directions of polarization of the two channels. Alternatively, an interferometer, such as a Mach-Zehnder interferometer can be used to combine two beams of the same polarization to produce amplitude modulation (see, for example, Rod. C. Alferness, "Waveguide electro-optic modulators", IEEE transactions on microwave theory and techniques, Vol. MTT-30, pp. 1121-1137, 1982). In such a device, the two channels of propagation are physically distinct waveguides.
In the linear electrooptic modulators, for each of the channels, the relation of phase velocity to applied voltage depends on the direction of the associated electric field produced in the modulator. The phase shift is proportional to the magnitude of the electric field and to the length L of the light path through the modulator. When the applied electric field is parallel to the direction of transmission, the amount of phase shift is independent of the length for a given applied voltage. An applied field perpendicular to the direction of transmission is advantageous because the electrodes do not then interfere with the propagation of the optical beam and because the amount of modulation, for a given applied voltage, can be increased by increasing the length of the crystal.
For modulation frequencies high enough that the transit time of the optical beam through the crystal is on the order of or greater than the period of the modulator frequency, the amount of modulation is proportional to the time integral of the applied signal over the transit time of the beam. Over such transit time, negative values of the applied voltage will offset the effects of positive values. In order to avoid such cancellation, the voltage is applied as a travelling wave that travels in the same direction as the optical beam. If the velocity of the travelling wave applied voltage equals the velocity of the optical beam in the modulator, then a given segment of the optical beam is subjected to a constant applied electric field as it travels through the modulator.
Unfortunately, the group velocity of the applied voltage signal is generally not equal to the group velocity of the light in the modulator. This results because the group velocity (in the absence of dispersion) is equal to the speed of light c divided by the index of refraction of the medium and because the index of refraction for the frequencies of the applied voltage is different from the index of refraction for the frequencies of the optical signal. For example, in LiNbO.sub.3 the index of refraction for an rf applied voltage is on the order of 4 whereas the index of refraction for optical frequencies is on the order of 2. As a result of this, a given segment of the optical beam does not experience a constant applied electric field. The effect of this can be easily seen for an optical signal EQU V.sub.o =A.sub.o *e.sup.i(w.sbsp.o.sup.t-k.sbsp.o.sup.z) ( 1)
having its phase modulated by an applied voltage EQU V.sub.a =A.sub.a *e.sup.i(w.sbsp.a.sup.t-k.sbsp.a.sup.z) ( 2)
The z axis has been chosen to lie along the direction of propagation of these two travelling waves and the point z=0 has been chosen to be at the input end of the modulator. The phase velocities of the optical beam and the applied voltage signal are v.sub.o =w.sub.o /k.sub.o and v.sub.a =w.sub.a /k.sub.a, respectively. The portion of the optical beam that enters the modulator at time t is located at EQU z=z.sub.o (t')=v.sub.o *(t'-t) (3)
at time t'. This portion of the optical field experiences at the point (t',z(t')) a retardation proportional to the applied field at the point--namely EQU V.sub.a (t',z.sub.o (t'))=A.sub.a *e.sup.i[w.sbsp.a.sup.t'-k.sbsp.a.sup.*v.sbsp.o.sup.*(t'-t)]( 4)
The total phase shift on this portion of the wave is equal to the time integral over t'-t from t'-t=0 to t.sub.o where t.sub.o is the transit time for the optical beam to cross the modulator and is equal to L*k.sub.o /w.sub.o. The effect of this is that the retardation is reduced by the factor EQU [e.sup.i(w.sbsp.r.sup.t.sbsp.o.sup.) -1]/iw.sub.r t.sub.o =e.sup.i(w.sbsp.r.sup.t.sbsp.o.sup./2) *sinc(w.sub.r t.sub.o /2)(5a) EQU where
w.sub.r =w.sub.a -v.sub.o *k.sub.a =w.sub.a *(1-v.sub.o /v.sub.a)(5b)
compared to the retardation that would result if the velocities v.sub.o and v.sub.a were equal. This walkoff of the phase of the applied voltage signal relative to the phase of the optical signal thus produces a reduction factor that is dependent on the frequencies of both signals.
The sinc function first goes to zero when its argument w.sub.r */t.sub.o /2 equals .+-.pi. Using equation (5a), the first null occurs when w.sub.a =2pi/(t.sub.a -t.sub.o)=2pi/(L/v.sub.a -L/v.sub.o), where t.sub.a is the transit time for the microwave to cross the modulator. This shows that the bandwidth varies inversely with L. This means that the bandwidth can be increased by decreasing the length of the modulator. Unfortunately, reducing the length of the modulator equivalently reduces the time during which the applied voltage affects the optical signal so that the magnitude of the modulation varies inversely with the length L of the region of modulation. Therefore, in the variation of the length L, there is a tradeoff between the bandwidth and the magnitude of the applied voltage required to produce a given amount of phase change. A measure of the applied voltage needed in the modulator is the voltage V.sub.pi which is defined to be the value of the dc voltage needed to produce a phase change of pi in the output optical signal. The ratio of bandwidth (BW) and V.sub.pi is a figure of merit that is independent of the length of the modulation region. This bandwidth-voltage-ratio (BVR) is thus a useful figure of merit of the modulators.
In one technique of increasing the upper limit of the useful band of applied frequencies (see Rod. C. Alferness, et al, "Velocity-matching techniques for integrated optic travelling wave switch/modulators", IEEE J. Quant. Electron, vol. QE-20, pp. 301-309, 1984), the electrodes have a shape that periodically reverses the applied electric field in the modulator as a function of z. Such periodic field reversals are used to offset the negative portions of the relative phase between the applied signal and the optical signal. Unfortunately, this cancellation is complete only at one value of w.sub.r, and, in addition, these periodic filed reversals degrade the low frequency performance. In effect, these periodic field reversals serve to shift the effective band upward in frequency without broadening the width of the band.
In another modulator (see A. Djupsjobacka, "Novel type of broadband travelling-wave integrated-optic modulator", Electronics Letters, pp. 908-909, 1985) there is only a single phase reversal produced by laterally offsetting the electrodes three-fourths of the distance along the modulator. It is asserted incorrectly that this design acts like a low pass filter and a high pass filter in series, whereas in fact it functions as a low pass filter and a high pass filter in parallel. Unfortunately, the increase in bandwidth with this structure is offset by a voltage reduction factor of 2. Thus, this device exhibits a reduced bandwidth-voltage-ratio (BVR) relative to a conventional Mach-Zehnder modulator having no polarity reversals. It would be useful to have a design that increases the bandwidth-voltage-ratio (BVR) and also retains a low value of V.sub.pi down to dc applied voltages.